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REVIEW 4 major objections 6 minor 39 references

Linear and Second-order-cone Valid Inequalities for Problems with Storage

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Pith's one-line read A submodularity structure in a battery's extreme power trajectories yields facet-defining linear valid inequalities for its convex hull, plus second-order-cone cuts that nearly eliminate simultaneous charge/discharge in two storage…

desk verdict A novel submodularity-based construction of storage cuts with a clean SOC convex hull, but the facet-defining claim is not proven as written and needs a real dimension argument before the paper's strongest claims can stand. read the letter →

arxiv 2506.21470 v1 pith:HA5DOX4T submitted 2025-06-26 math.OC

Juan M. Morales This is my paper · ORCID
classification math.OC MSC 90C1190C5790C2590C26
keywords energystoragesystemsbatterymodelingcomplementarityconstraintssubmodularityvalidinequalitiesconvexhullsecond-orderconeprogrammingmixed-integer
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

To model a battery you must forbid charging and discharging in the same time period, and the usual complementarity constraint makes the feasible set disjunctive and the resulting optimization problems NP-hard. This paper claims that the corner points of that feasible set are governed by a submodularity property, and that this property can be converted into a family of linear inequalities that are valid for the exact nonconvex set and define facets of its convex hull. If correct, this gives modelers a way to tighten the ordinary linear relaxation of storage problems without adding binary variables, and one member of the family generalizes and dominates the existing single-period facet cuts. For the set-point tracking problem the paper adds second-order-cone inequalities and reports reductions of 99.50% in simultaneous charge/discharge hours and 99.96% in average violation magnitude across 20,000 instances.

What carries the argument

The load-bearing object is the set function $f(\Omega)$, defined in (5) as the maximum cumulative charging power over a horizon when discharging is allowed only in the periods $\Omega$ and charging is forced elsewhere. Proposition 1 establishes that $f$ is submodular: the marginal gain of adding a discharge period is no larger in a bigger context. Submodularity, together with the complement-function lemma (a submodular function's complement is submodular), produces inequality (13), which becomes the linear cut (10) when gains $f(\tau|\emptyset)$ are converted into coefficients $\rho_c$ with a sign-dependent scaling by discharge power limits. For the set-point tracking problem the central object is the per-period epigraph $E$; Theorem 2 shows its convex hull is the epigraph of a parabolic cylinder, which is second-order-cone representable, and the cross-term $2p^dp^c$ in the cylinder equation is exactly what penalizes simultaneous charge and discharge.

What would settle it

Enumerate all $2^T$ subsets $\Omega$ for a small horizon, say $T=4$, with fixed efficiencies and power and energy limits; compute $f(\Omega)$ by brute force and check every submodularity inequality $f(\{j\}|A) \ge f(\{j\}|B)$ for $A \subseteq B$ and $j \notin B$. A single violation would falsify Proposition 1 and hence the validity of (10) and (11). A companion check is to verify that the definition of $\Omega$ in (6e) and (6f) matches the complement-function usage in Theorem 1, since a reversed convention would break the proof.

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Extended reading notes

Core claim

The central claim is that the linear inequalities (10) and (11) are valid for the feasible set $P$ and define facets of $\operatorname{conv}(P)$. Inequality (10) bounds a sum of charging powers $p^c_{t+\tau}$ by a baseline $\sum_\tau c(t,\tau)$ plus a weighted sum of discharging powers $p^d_{t+\tau}$, with coefficients $\rho_c(t,\tau,\bar\tau)$ computed from the gains of the submodular function; (11) is the symmetric discharging inequality. Corollary 2 asserts that this multi-period family dominates and generalizes the single-period facet inequalities of [27]. The argument runs through submodularity of the maximum-charge function $f(\Omega)$ in (5) and of its complement, which supplies the upper bound used to construct the cuts. The second-order-cone contribution is Theorem 2: for the set-point tracking problem, the convex hull of the per-period disjunctive set $E$ is the epigraph of the parabolic cylinder $(p^d+p^c)^2 - 2(p^d-p^c)p^s + (p^s)^2 \le z$, giving a conic valid inequality that nearly eliminates simultaneous charge/discharge in the reported experiments.

Load-bearing premise

The load-bearing premise is that the maximum charging power obtainable by choosing which periods may discharge is submodular: adding a discharge period to a larger set of allowed discharge periods yields no more extra charge than adding it to a smaller set. The paper proves this by a recursive exchange argument, and the validity of inequalities (10) and (11) collapses if that argument or the complement-function step in Theorem 1 is wrong.

Editorial extensions

If this is right

  • Storage models can be tightened by adding the multi-period linear cuts (10) and (11) to a continuous relaxation instead of binary complementarity variables, and Proposition 2 guarantees those cuts are facet-defining rather than merely valid.
  • The new inequalities dominate the single-period facet cuts of [27], so models already using those cuts can replace them without losing strength.
  • The $u$-variable inequalities (21) and (22) offer a complementary way to tighten the mixed-integer formulation, giving modelers a choice between power-space and binary-space cuts depending on which is stronger.
  • For the set-point tracking problem, the SOC cut (28) can be added to the continuous relaxation and yields nearly exact physical behavior: in the reported 20,000 instances, simultaneous charge/discharge hours drop from 15.76% to 0.08%.
  • Because the cuts are valid for the exact feasible set, any relaxed solution that still shows simultaneous charge/discharge indicates the relaxation remains inexact, which is a useful diagnostic for further strengthening.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Outside the paper, one would expect the submodularity exchange argument to extend to more general state-of-charge dynamics, including efficiency that varies with power, since the proof only uses monotonicity of the state of charge in the allowed discharge set.
  • A separation algorithm for the exponential family (10) and (11) is the natural next step; the paper's own redundancy analysis already suggests that selective cut generation could recover most of the solution-time speed-up that the full constraint set sacrifices.
  • The parabolic-cylinder construction is per-period, so combining it with multi-period state coupling or with objectives that penalize charge-discharge transitions is a testable extension the paper does not explore.
  • If the facet claims hold, the convex hull $\operatorname{conv}(P)$ of the storage set may be describable by these families, which would imply that the NP-hardness of storage scheduling comes from the objective and integrality elsewhere, not from the storage feasible set itself.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper studies the nonconvex feasible operating set P of a battery with complementarity constraints, together with its mixed-integer formulation and linear relaxation. It claims that extreme power trajectories exhibit a submodular structure, and it uses this structure to construct families of linear valid inequalities (10)-(11). The paper further claims that these inequalities define facets of conv(P) and, as a consequence, generalize and dominate the single-period facet inequalities of Pozo [27]. For the set-point tracking problem, it derives a second-order-cone valid inequality (28), and it reports numerical experiments on 1000 arbitrage instances and 20,000 tracking instances showing substantial reductions in simultaneous charging and discharging.

Significance. If the submodularity-based construction and the facet claim are fully established, the paper would provide a systematic, parameter-free way to strengthen linear relaxations of multi-period storage models and would improve on existing single-period facet inequalities. The construction is self-contained and not fitted to the data, and the computational study is substantial and reproducible through the public repository. The reported reductions in complementarity violations are concrete and practically relevant. However, the facet proof is currently not valid, so the main theoretical contribution is conditional on a repair; the numerical value of the inequalities as valid cuts is less affected by that gap.

major comments (4)
  1. [III, Proposition 2 and Corollary 2] The proof of Proposition 2 does not establish that (10)-(11) are facet-defining. It only shows that (14) is tight at vertices corresponding to B=∅ and B={j}, j∈T, and then asserts that 2|T| tight vertices imply a facet. This inference is invalid: a face of conv(P) can contain many vertices without having affine dimension dim(conv(P))-1, and a valid inequality can be tight on many vertices without being a facet. Since Corollary 2 explicitly derives the claimed dominance and generalization from the facet statement, the paper's central theoretical claim is unproved as written. The authors should replace this argument with an affine-independence or face-dimension computation, or weaken the claims to validity of the inequalities.
  2. [III, Proposition 1 and Theorem 1] The definition of the set function f in (5)-(6) is inconsistent with the role it plays in Theorem 1. In (6e)-(6f), Ω is the set of periods in which discharge is forbidden and charging is permitted, so f(Ω) is the maximum cumulative charge when only periods in Ω may charge. However, the proof of Theorem 1 treats B as the set of discharging periods and identifies f(∅) with the cumulative charge when no discharge is allowed; this identification holds only for the complement function f(A)=f(T\A), not for f as defined. The notation for the original and complement functions is not kept distinct, which makes the derivation of (14) and the interpretation of f(τ|∅) impossible to verify as written. Please restate the definitions with the original and complement functions clearly separated.
  3. [III, Theorem 1, definitions of ρc and ρd] The displayed formulas for ρc(t,τ,τ) and ρd(t,τ,τ) are self-referential: the same symbol appears on both sides of the definition, since the condition 'if ρc(t,τ,τ) ≤ 0' uses the quantity being defined. The subsequent sentence in the proof, ρc(t,τ,τ)=ρc(t,τ,τ)/δ_{t+τ}, also appears to confuse the raw gain with the normalized coefficient. Because (10)-(11) are the central objects of the paper, these definitions need to be written with distinct symbols and a clear derivation.
  4. [Appendix A, proof of Proposition 1] The submodularity proof is only a sketch at several key points: the exchange argument is summarized as 'we can recursively use these strategies,' and boundary cases such as Ω=∅, Ω=T, j=τ, and states at S or S are not treated explicitly. Since Proposition 1 is the foundation of Theorem 1 and of the claimed facet result, the proof should be completed or replaced by a fully formal argument. In particular, the inequality f(τ|Ω≤τ−1∪{j}) ≤ f(τ|Ω≤τ−1) is asserted for j<τ and j>τ with different informal justifications; a unified proof is needed.
minor comments (6)
  1. [IV, Equation (28)] The displayed expression for conv(E) is garbled in the current typesetting, with the matrix-vector terms not rendered readably. Please display the equivalent explicit form z_t ≥ (p^d_t+p^c_t)^2 - 2p^s_t(p^d_t-p^c_t) + (p^s_t)^2, which is what the geometric proof actually establishes.
  2. [References and Section V] Reference [32] contains a space in the URL and is not clickable as printed; also, the sentence 'All the data and codes necessary for reproducing the results of this article available from' is missing a verb.
  3. [III, Remark 2] Remark 2 uses the symbol E(t) in the definition of Γc(t) although the quantity defined just above is Ec(t); please make the notation consistent.
  4. [III, Corollary 2] The proof of Corollary 2 invokes the facet claim, but the coefficient-wise comparison in the second paragraph of the proof is sufficient to show dominance. That comparison should be presented as the proof, independently of Proposition 2.
  5. [V, experimental setup] The numerical section states that TLP uses 'the valid inequalities (10) and (11)' but does not specify whether the redundant constraints identified in Remark 2 are removed or kept. Please state the exact set of constraints used in the experiments, since the reported timings and tightness can depend on this choice.
  6. [IV, proof of Theorem 2] In the convex-combination argument in the proof of Theorem 2, the case p^d+p^c=0 is not separately handled, since the expression λ = p^d/(p^d+p^c) is undefined there; this is a trivial boundary case but should be mentioned.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the facet-proof gap in Proposition 2 is a correctness issue, not a circular derivation, and the only self-reference is a non-load-bearing data/code repository.

full rationale

The derivation chain is self-contained. Inequalities (10)-(11) are constructed from the submodularity of f in (5), proven in Appendix A, and their validity is shown via the submodular-complement inequality (12)-(14); the coefficients rho_c and rho_d are computed analytically from the extreme state trajectories (8)-(9), with no parameter fitted to the numerical experiments. The dominance over Pozo [27] is justified by explicit coefficient-wise comparisons in Corollary 2. The SOC inequalities (28) are derived from an explicit convex-hull computation of the two-curve disjunctive set E, with containment verified in Theorem 2. The main caveat is Proposition 2, whose proof says: 'Therefore, in general (excluding the redundant constraints indicated in Remark 2), (14) is satisfied with equality at 2|T| distinct vertices, which implies that it defines a facet of the polytope conv(P).' This is a missing affine-dimension argument and a correctness risk, not circularity, because the facet claim is not an input to the construction of the inequalities; Corollary 2 inherits the gap but does not feed back into the derivation. The only self-reference is the OASYS GitHub repository [32], cited for data/code reproduction and not load-bearing. Thus no circular step reduces the paper's results to its inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted. The derivation uses standard submodularity and convex analysis, plus the domain assumption that the standard battery model is correct. No new physical entities are introduced.

assumptions (5)
  • domain assumption The battery feasibility set P in (1), including complementarity, is the correct operational model.
    The entire derivation is for this set; any deviation (e.g., degradation, SoC-dependent efficiency) is outside scope.
  • domain assumption The effective maximum charge and discharge rates P_c^e and P_d^e in (4) can replace P_c and P_d without changing the feasible set.
    Used to tighten the relaxation; standard result but relied upon in all derivations.
  • standard math The value function f in (5) is submodular, and its complement is submodular (Lemma 1).
    Proven in Appendix A, but the proof's exchange argument is the core technical dependency for (10)-(11).
  • standard math Submodularity implies the bound (13): f(B) <= f(empty) + sum_{tau in B} f(tau|empty).
    This is a standard property of submodular functions, but the paper uses it to construct the inequalities.
  • standard math The convex hull of E in (25) is exactly the epigraph of the parabolic cylinder with lambda = 2.
    Proven in Theorem 2 using a pencil-of-quadrics construction; this assumption is more self-contained but still relies on the geometric containment proof.

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Pith. "Pith review of Linear and Second-order-cone Valid Inequalities for Problems with Storage." pith.science (2026). https://pith.science/paper/HA5DOX4T

@misc{pith2026250621470,
  author       = {Pith},
  title        = {Pith review of: Linear and Second-order-cone Valid Inequalities for Problems with Storage},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HA5DOX4T}},
  note         = {Machine review of arXiv:2506.21470}
}
read the original abstract

Batteries are playing an increasingly central role as distributed energy resources in the shift toward power systems dominated by renewable energy sources. However, existing battery models must invariably rely on complementarity constraints to prevent simultaneous charging and discharging, rendering models of a disjunctive nature and NP-hard. In this paper, we analyze the disjunctive structure of the battery's feasible operational set and uncover a submodularity property in its extreme power trajectories. Leveraging this structure, we propose a systematic approach to derive linear valid inequalities that define facets of the convex hull of the battery's feasible operational set, including a distinguished family that generalizes and dominates existing formulations in the literature. To evaluate the practical utility of these inequalities, we conduct computational experiments on two representative problems whose continuous relaxations frequently result in simultaneous charge and discharge: energy arbitrage under negative prices and set-point tracking. For the latter, we further introduce second-order cone inequalities that remarkably and efficiently reduce simultaneous charging and discharging. Our results highlight the potential of structured relaxations to enhance the tractability and fidelity of battery models in optimization-based energy system tools.

Figures

Figures reproduced from arXiv: 2506.21470 by the authors.

Figure 1
Figure 1. Illustration of the parabolic cylinder C for p s = 1. Now we verify that the desired (parabolic) cylinder C is the member of the family that corresponds to λ = 2 and satisfies the equation z = (p d ) 2 + (p c ) 2 + 2p d p c − 2p d p s + 2p c p s + (p s ) 2 Here we observe the term 2p dp c , which will effectively penalize the simultaneous charge and discharge of the ESS. Moreover, the epigraph of this cylinder is se… view at source ↗

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